Can someone verify the calculation?
the calculation is generally correct, but dependent on your definition of "turn on". for most analog applications, people use "turn on" to mean "starting to turn on". and for digital applications, people tend to use "fully turn on" instead.
as Vbe is not a constant, nor the reverse voltage of the zener, the figure you obtained is just a rough estimate, and should be adjusted further for specific applications if called for.
How will calculation changes if drop across R1 also need incorporate?
you should calculate the Ic first, and then apply a Ib based on the assumed beta of the transistor. That current will cause a voltage drop over R1.
If you use typical values, you will find that that voltage drop is quite small vs. Vbe + Vzfwd, thus most times ignored.
However, there can be cases where the voltage drop over R1 is significant. A prudent design practice is to first assume that it is insignificant, go through your calculations, obtain Ib and then confirm that it is indeed significant.
I'm amending this in response to the answer provided by Jack Creasey below. Jack had concluded that the Ib would be 16ua max, based on an estimated beta of 50.
I pointed out that beta is really a concept for linear applications (where the associated Vce is much higher than in a switching application here).
Here is a simulation I put together quickly - it mimics the circuit provided by the OP, with the exception that I used a 4.7v zener, not a 2.7v zener. the use of a 2.7v zener would have increased the Ib by 200ua in this case, with minimum changes in Ic.
the blue line in the plot is Ib and it reaches 500ua at 10v input, with Ic at 800ua -> a beta of 1.6.
hope it helps.
if anyone is interested, I can simulate the case with a 2.7v zener, :)
edit2: due to popular demand, here is the same circuit, using a 2.7v zener.
Ib tops out at about 680ua, vs. 500ua with a 4.7v zener, Ic unchanged, just as we expected.
again, the point I have been trying to emphasize is that beta is a useful concept for linear applications and has no meaning in a switching application, as shown here.